MnO4- + 8H+ + 5e <-> Mn2+ + 4H2O
the oxidising power of potassium permanganate is governed by the above equation. The forward reaction is favoured (Le Chateliter's Principle) in an acidic medium.
The only reason why dilute sulphuric acid is the preferred acid medium is because not only it provides H+, but also HSO4- and SO42- are not easily oxidised/reduced to yield unwanted products.